Laboratory experiments on thin-walled tubes at large finite strain: Symmetry, coaxiality, rigid body rotation, and the role of invariants, for the applied stress σ = RTRT, the cauchy stress σ∗ = [IIIV]−1FTRT, and the left cauchy-green stretch tensor V = FRT

Abstract

From laboratory measurements extending back to the mid-1960s, a frame indifferent general continuum theory for large finite plastic strain in ordered solids has been developed. Its foundations are based on a stress σ that replaces the Cauchy stress σ∗ in the applicable constitutive statements. With W the work per unit volume in the underformed reference configuration, a work function W = traceσV, established from experiment, is consistent with an internal constraint on the left Cauchy-Green stretch tensor, trace V = 3, in whose presence simple shear becomes inadmissible. Compatible with this internal constraint, the decrease in volume during deformation is measured. Also shown in the laboratory is the fact that even when determined just prior to catastrophic collapse, the measured rigid body rotation is too small to play other than a negligible role. In this article, laboratory experimental results are presented which reveal that the Cauchy stress σ∗, the stress σ of the present theory, and the left Cauchy-Green stretch tensor V, are all coaxial and symmetric. Moreover, for the approximation R = I, the second invariant T of the deviatoric stress S and the second invariant Γ of the finite strain E = V − I provide a universal parabolic rule that applies for stress paths of arbitrary composition and direction. This general nonlinear theory of finite strain plasticity during loading is shown here to square in detail with a recently published general nonlinear theory of finite strain elasticity derived from the analytical foundations of rational mechanics by Beatty and Hayes [1992].